Question

The​ quality-control manager at a compact fluorescent light bulb​ (CFL) factory needs to determine whether the mean life of a large shipment of CFLs is equal to 7 comma 470 hours. The population standard deviation is 100 hours. A random sample of 64 light bulbs indicates a sample mean life of 7 comma 445 hours.

a. At the 0.05 level of​ significance, is there evidence that the mean life is different from 7 comma 470 hours question mark

b. Compute the​ p-value and interpret its meaning.

c. Construct a 95​% confidence interval estimate of the population mean life of the light bulbs.

d. Compare the results of​ (a) and​ (c). What conclusions do you​ reach?

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis: μ = 7470
Alternative Hypothesis: μ ≠ 7470

Rejection Region
This is two tailed test, for α = 0.05
Critical value of z are -1.96 and 1.96.
Hence reject H0 if z < -1.96 or z > 1.96

Test statistic,
z = (xbar - mu)/(sigma/sqrt(n))
z = (7445 - 7470)/(100/sqrt(64))
z = -2

Rejection Region Approach
As the value of test statistic, z is outside critical value range, reject the null hypothesis

There is sufficient evidence to conclude that the mean life is different from 7 comma 470 hours question

b)

P-value Approach
P-value = 0.0455
As P-value < 0.05, reject the null hypothesis.

c)

sample mean, xbar = 7445
sample standard deviation, σ = 100
sample size, n = 64

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

ME = zc * σ/sqrt(n)
ME = 1.96 * 100/sqrt(64)
ME = 24.5

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (7445 - 1.96 * 100/sqrt(64) , 7445 + 1.96 * 100/sqrt(64))
CI = (7420.5 , 7469.5)

d)

Results are same in A) and c)

Reject the null hypothesis